2017 AMC 12A Problems/Problem 12

Problem

There are $10$ horses, named Horse 1, Horse 2, $\ldots$ , Horse 10. They get their names from ow many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$ , in minutes, at which all $10$ horses will gain simultaneously be at the starting point is $S = 2520$ . Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$ ?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Solution

We know that Horse $k$ will be at the starting point after $n$ minutes iff $k|n$ . Thus, we are looking for the smallest $n$ such that at least $5$ of the numbers $\{1,2,\cdots,10\}$ divide $n$ . Thus, $n$ has at least $5$ positive integer divisors.

We quickly see that $12$ is the smallest number with at least $5$ positive integer divisors, and that $1,2,3,4,6$ are each numbers of horses. Thus, our answer is $1+2=\boxed{\textbf{(B) } 3}$ .

2017 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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All AMC 12 Problems and Solutions

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2017 AMC 12A Problems/Problem 12

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